Beyond Outerplanarity
Steven Chaplick, Myroslav Kryven, Giuseppe Liotta, Andre L\"offler,, Alexander Wolff

TL;DR
This paper investigates convex drawings of graphs, introducing outer k-planar and outer k-quasi-planar classes, providing structural properties, complexity results, and logical characterizations, with implications for graph coloring, separators, and recognition algorithms.
Contribution
It establishes new structural bounds, complexity results, and logical frameworks for outer k-planar and quasi-planar graphs, including coloring, separators, and linear-time recognition.
Findings
Outer k-planar graphs are -degenerate.
Every outer k-planar graph can be colored with +1 colors.
Outer k-planar graphs have small balanced vertex separators.
Abstract
We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., \emph{convex drawings}. We consider two families of graph classes with convex drawings: \emph{outer -planar} graphs, where each edge is crossed by at most other edges; and, \emph{outer -quasi-planar} graphs where no edges can mutually cross. We show that the outer -planar graphs are -degenerate, and consequently that every outer -planar graph can be colored with colors. We further show that every outer -planar graph has a balanced vertex separator of size at most . For each fixed , these small balanced separators allow us to test outer -planarity in quasi-polynomial time, e.g., this implies that none of these recognition problems is NP-hard unless the Exponential Time Hypothesis…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · Complexity and Algorithms in Graphs
